If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional analysis can be defined as the practice of studying relations between physical quantities by recognising the dimensions of them. These dimensions are independent of numerical multiples and similar constants. The quantities present in the world can be described as a function of the fundamental dimensions.

Dimensional Formula

Dimensional formula of a quantity is the expression indicating the powers wherein the fundamental units are to be increased to obtain one unit of a derived quantity. For instance, if Q is an unit of a derived quantity which is defined by Q = M^aL^bT^c, then M^aL^bT^c is known as the dimensional formula with exponents a, b and, c that are called the dimensions.

Dimensional Formula for Resistance

The dimensional formula of resistance can be expressed as:

M¹L²T^-3I^-2

Here,

M = Mass
I = Current
L = Length
T = Time

Resistance (R) = Voltage × Current^-1… [1]

Now, because, Voltage (V) = Electric Field × Distance = [Force × Charge^-1] × Distance

Accordingly, dimensional formula of charge = current × time = I¹T¹

Dimensional formula of voltage = [Force × Charge^-1] × Distance

= [M¹ L¹ T^-2] × [I¹T¹]^-1 × [L¹]

= [M¹ L² T^-3I^-1] … [2]

On replacing equation (ii) in equation (i) we get,

Resistance (R) = Voltage × Current^-1

R = [M¹ L² T^-3I^-1] × [I]^-1 = [M¹ L² T^-3I^-2]

Hence, resistance is dimensionally depicted as M L² T^-3I^-2.

Dimensional Formula of Capacitance

The dimensional formula of Capacitance is expressed by:

M^-1L^-2T⁴I²

Here,

M = Mass
I = Current
L = Length
T = Time

Derivation of Capacitance

Capacitance (C) = Charge × Voltage^-1. . . [1]

Since, Charge = Current × Time

∴ Therefore, the dimensional formula of charge can be give as = [I¹T¹] . . . . [2]

And, Voltage = Electric Field × Distance . . [3]

Now, as per Electric Field,

Electric Field = [Force × Charge^-1]

Hence, dimensional formula of force and charge equals [M¹ L¹ T^-2] and [I¹T¹] respectively.

∴ The dimensional formula of Electric Field = [M¹ L¹ T^-2] × [I¹T¹]^-1

= [M¹ L¹ T^-3 I^-1] . . . [4]

On replacing equation [4] in equation [3] we can obtain,

The dimensional formula of Voltage = [M¹ L¹ T^-3 I^-1] × [L¹]

= [M¹ L² T^-3I^-1] . . . [5]

On replacing equations [5] and [2] in equation [1] we can obtain,

Hence, Capacitance = Charge × Voltage^-1

Or, as can be expressed, C = [I¹T¹] × [M¹ L² T^-3I^-1]^-1 = [M^-1 L^-2 T⁴I²]

Thus, the Capacitance can be dimensionally expressed as [M^-1 L^-2 T⁴I²].

Topics of Resistance
Potentiometer	Conductivity	Ampere
Cells, emf, Internal Resistance	Unit of Specific Resistance	Power of Alternating Current
Combination of Resistors	Cells in Series and in Parallel	Ohm’s law
Wheatstone Bridge	Internal resistance formula	Resistivity Temperature Dependence
Types of Current	Difference between air conditioning and refrigeration	Permittivity And Permeability
Types of Resistors	Derivation of Drift Velocity	Power transformers
Difference between kva and kw	Difference between RTD and thermocouple	Difference between AC and DC
Ohm’s Law and its Limitations	Relation between kva and kw	Static Electricity
Voltage	Electrical Formula	Resistor Colour Codes
Current density	Difference between emf and voltage	Resistivity formula
Current density formula	Relation between resistance and length	Unit of Current
Electric Dipole	Difference Between Primary Cell and Secondary Cell	Electromotive Force
Current Electricity	Resistance Formula	Current Electricity MCQ
Carbon Resistor	Differences Between Acceleration and Velocity	Relation between power and resistance
Uses of Resistor	Difference between earth and neutral	Drift Velocity

1.
Read the following paragraph and answer the questions that follow.
A p-type or n-type semiconductor can be converted into a p-n junction by doping it with suitable impurity. The motion of majority charge carriers causes diffusion current across the junction while the barrier electric field causes motion of minority carriers for drift current. In case of unbiased diode, the diffusion and drift currents are equal. This equilibrium is disturbed by the biasing batteries. Diodes, therefore, allow currents in one direction. This property of diode is used in making rectifiers.

2.
A charged particle $+q$ in an electric field $\vec{E}$ experiences a force in the direction of the electric field. As a result, its kinetic energy changes. Similarly, the charged particle also experiences a force when it moves in a magnetic field $\vec{B}$. But this magnetic force is perpendicular to both velocity $\vec{v}$ of the charged particle and the magnetic field $\vec{B}$, so it cannot change the kinetic energy of the charged particle. Consider two charged particles 1 and 2 of masses $m$ and $\frac{m}{2}$ having charges $-q$ and $+2q$ respectively. They are accelerated from rest through the same potential difference $V$ and acquire kinetic energy $K_1$ and $K_2$. Then they enter in a region of uniform magnetic field $\vec{B}$ perpendicular to their velocities.

3.
Two metal spheres of radii $r_1$ and $r_2$ ($> r_1$) having charges $q_1$ and $q_2$ respectively kept in air, are brought in contact. Which of the following statements is not correct ?

The total charge of the two spheres is conserved.
Both spheres attain the same potential.
The final potential of the system equals $\frac{1}{4\pi\epsilon_0} \frac{(q_1 + q_2)}{(r_1 + r_2)}$
The final potential of the system equals $\frac{1}{4\pi\epsilon_0} \frac{(q_1 + q_2) (r_1 + r_2)}{r_1 r_2}$

View Solution

4.
Consider the nuclear reaction $ X \to Y + Z $. Let $ M_x $, $ M_y $, and $ M_z $ be the masses of the three nuclei X, Y, and Z respectively. Then which of the following relations hold true?

$ (M_x - M_z)<M_y $
$ (M_x - M_y)<M_z $
$ M_x>(M_y + M_z) $
$ M_x<(M_y + M_z) $

View Solution

5.
The resistance of a metal wire at $ 20^\circ \text{C} $ is $ 1.05 \, \Omega $ and at $ 100^\circ \text{C} $ is $ 1.38 \, \Omega $. Determine the temperature coefficient of resistivity of this metal.

View Solution

6.
Read the following paragraph and answer the questions that follow.
In an experiment with convex lens of focal length f, the screen is fixed at a distance D from the object. A student slowly moves the lens away from the object towards the screen and finds that she is able to form sharp image of the object for two positions of the lens. The distance between these two positions of the lens is d.

If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional Formula

Dimensional Formula for Resistance

Dimensional Formula of Capacitance

Derivation of Capacitance

Related Questions

CBSE CLASS XII Related Questions

3.
Two metal spheres of radii $r_1$ and $r_2$ ($> r_1$) having charges $q_1$ and $q_2$ respectively kept in air, are brought in contact. Which of the following statements is not correct ?

4.
Consider the nuclear reaction \( X \to Y + Z \). Let \( M_x \), \( M_y \), and \( M_z \) be the masses of the three nuclei X, Y, and Z respectively. Then which of the following relations hold true?

5.
The resistance of a metal wire at \( 20^\circ \text{C} \) is \( 1.05 \, \Omega \) and at \( 100^\circ \text{C} \) is \( 1.38 \, \Omega \). Determine the temperature coefficient of resistivity of this metal.

Similar Physics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional Formula

Dimensional Formula for Resistance

Dimensional Formula of Capacitance

Derivation of Capacitance

Related Questions

CBSE CLASS XII Related Questions

3.Two metal spheres of radii $r_1$ and $r_2$ ($> r_1$) having charges $q_1$ and $q_2$ respectively kept in air, are brought in contact. Which of the following statements is not correct ?

4.Consider the nuclear reaction \( X \to Y + Z \). Let \( M_x \), \( M_y \), and \( M_z \) be the masses of the three nuclei X, Y, and Z respectively. Then which of the following relations hold true?

5. The resistance of a metal wire at \( 20^\circ \text{C} \) is \( 1.05 \, \Omega \) and at \( 100^\circ \text{C} \) is \( 1.38 \, \Omega \). Determine the temperature coefficient of resistivity of this metal.

Similar Physics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

3.
Two metal spheres of radii $r_1$ and $r_2$ ($> r_1$) having charges $q_1$ and $q_2$ respectively kept in air, are brought in contact. Which of the following statements is not correct ?

4.
Consider the nuclear reaction \( X \to Y + Z \). Let \( M_x \), \( M_y \), and \( M_z \) be the masses of the three nuclei X, Y, and Z respectively. Then which of the following relations hold true?

5.
The resistance of a metal wire at \( 20^\circ \text{C} \) is \( 1.05 \, \Omega \) and at \( 100^\circ \text{C} \) is \( 1.38 \, \Omega \). Determine the temperature coefficient of resistivity of this metal.